Observations Concerning the probability of the existence of annihilators for balanced boolean functions

TL;DR

This paper provides an almost exact estimate of the probability that balanced boolean functions have low-degree annihilators, aiding in designing cryptographic functions resistant to algebraic attacks amid increasing computational power.

Contribution

It introduces a novel, highly accurate method to estimate the probability of annihilators in balanced boolean functions, improving upon previous upper bounds.

Findings

The probability of low-degree annihilators can be estimated with high accuracy.

The new estimate significantly improves over previous upper bounds.

Results help in designing cryptographic functions resistant to algebraic attacks.

Abstract

LFSR-based stream ciphers with nonlinear filters or combiners are susceptible to algebraic attacks using linearization methods to solve an overdefined system of nonlinear equations. And this process is greatly enhanced if the filtering or combining function has a low degree annihilator. To prevent such an attack, one would choose the parameters of that function so that the degree of its annihilator becomes large enough. As computing power is continuously increasing, a choice that seems secure today, becomes insecure tomorrow. Therefore, a tool is needed to estimate the probability of the existence of annihilators for balanced boolean functions with parameters that are beyond the current computing power. Based on experimental and calculational observations, we give in this paper an almost exact estimate of that probability, which represent a great improvement over the upper bound previously known.